Solve for $x$ : $ 7|x + 4| + 7 = 5|x + 4| + 9 $
Answer: Subtract $ {5|x + 4|} $ from both sides: $ \begin{eqnarray} 7|x + 4| + 7 &=& 5|x + 4| + 9 \\ \\ { - 5|x + 4|} && { - 5|x + 4|} \\ \\ 2|x + 4| + 7 &=& 9 \end{eqnarray} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} 2|x + 4| + 7 &=& 9 \\ \\ { - 7} &=& { - 7} \\ \\ 2|x + 4| &=& 2 \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{2|x + 4|} {{2}} = \dfrac{2} {{2}} $ Simplify: $ |x + 4| = 1$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 4 = -1 $ or $ x + 4 = 1 $ Solve for the solution where $x + 4$ is negative: $ x + 4 = -1 $ Subtract ${4}$ from both sides: $ \begin{eqnarray} x + 4 &=& -1 \\ \\ {- 4} && {- 4} \\ \\ x &=& -1 - 4 \end{eqnarray} $ $ x = -5 $ Then calculate the solution where $x + 4$ is positive: $ x + 4 = 1 $ Subtract ${4}$ from both sides: $ \begin{eqnarray} x + 4 &=& 1 \\ \\ {- 4} && {- 4} \\ \\ x &=& 1 - 4 \end{eqnarray} $ $ x = -3 $ Thus, the correct answer is $x = -5 $ or $x = -3 $.